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CDC Class 8/9/10 Math Factorization Exercise 10.1.2 Solutions & Comprehensive Rules Nepal

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CDC Math Ex 10.1.2 Factorization

CDC Class 8/9/10 Math Nepal Solutions

Ex 10.1.1

Share CDC Math Ex 10.1.2 Solutions:

CDC Math Factorization Exercise 10.1.2 Video Class Guide

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Factorization Ex 10.1.2 Key Formulas & Principles

Always factor out any common terms first, then apply the difference of two squares formula $a^2 – b^2 = (a – b)(a + b)$ for CDC Class 8, 9, 10 Factorization Exercise 10.1.2.

खण्डीकरण गर्दा सम्भव भएसम्म पहिला साझा (Common) लिनुहोस् र त्यसपछि $a^2 – b^2$ को सूत्र प्रयोग गर्नुहोस्।

Difference of Squares Formula

$$a^2 – b^2 = (a – b)(a + b)$$

Common Factor Rule for Ex 10.1.2

$$ka^2 – kb^2 = k(a^2 – b^2)$$

Total Exercise Problems: 25

1. Factorize the following algebraic expressions (CDC Class 8/9/10 Math Ex 10.1.2):

१. दिइएका बीजीय अभिव्यञ्जकहरूलाई सूत्रको प्रयोग गरी खण्डीकरण गर्नुहोस्:

1.a

$$x^2 – 4$$

1.b

$$a^2 – 4b^2$$

1.c

$$9x^2 – y^2$$

1.d

$$5x^2 – 20y^2$$

1.e

$$13a^2 – 117b^2$$

1.f

$$25 – \frac{1}{9y^2}$$

1.g

$$121x^2 – \frac{1}{y^2}$$

1.h

$$2p^2 – \frac{50}{q^2}$$

1.i

$$72 – 2b^2$$

(e प्रश्न जस्तै समान प्रकृतिको / Same type as question e)

1.j

$$121 – 25y^2$$

1.k

$$\frac{15}{a^2} – 60a^2$$

1.l

$$81 – 64y^2$$

(j प्रश्न जस्तै समान प्रकृतिको / Same type as question j)

1.m

$$4x^3y – 81xy^3$$

1.n

$$169 – 196z^2$$

1.o

$$ab^3 – 9a^3b$$

(m प्रश्न जस्तै समान प्रकृतिको / Same type as question m)

1.p

$$\frac{49}{121}x^2 – \frac{64}{9}y^2$$

1.q

$$zx^2 – zy^2$$

(m प्रश्न जस्तै समान प्रकृतिको / Same type as question m)

1.r

$$(x+2)^2 – 4$$

1.s

$$256 – \frac{x^2}{4}$$

(k / f प्रश्न जस्तै समान प्रकृतिको / Same type as question k / f)

1.t

$$1 – \frac{81p^2}{121q^2}$$

1.u

$$3(x – y)^2 – 12$$

1.v

$$9(x-1)^2 – 16(x+2)^2$$

What is the area of shaded part in the given figure? (CDC Math Factorization Ex 10.1.2)

दिइएको चित्रमा छाया पारिएको भागको क्षेत्रफल कति हुन्छ?

Figure: Square with outer side $x\text{ cm}$ and inner side $3\text{ cm}$

Given Parameters:

Length of outer square ($L$) = $x\text{ cm}$
Length of inner smaller square ($l$) = $3\text{ cm}$

सुझाव: छाया पारिएको भागको क्षेत्रफल = (ठूलो वर्गको क्षेत्रफल) – (सानो वर्गको क्षेत्रफल)

There is a square shaped pond of length $6\text{ m}$ inside a square shaped garden of length $x\text{ m}$. What will be the area of garden excluding the pond? (CDC Math Ex 10.1.2)

$x\text{ m}$ लम्बाइ भएको वर्गाकार बगैँचाभित्र $6\text{ m}$ लम्बाइ भएको वर्गाकार पोखरी छ। पोखरीबाहेक बगैँचाको क्षेत्रफल कति होला?

Figure: Square Garden ($x\text{ m}$) with Inner Pond ($6\text{ m}$)

Given Information:

Length of square garden ($L$) = $x\text{ m}$
Length of square pond ($l$) = $6\text{ m}$

बगैँचाको सम्पूर्ण क्षेत्रफलबाट पोखरीको क्षेत्रफल घटाउँदा पोखरीबाहेकको बगैँचाको बाँकी भाग प्राप्त हुन्छ।

Observe the given figure and answer the following questions (CDC Math Ex 10.1.2):

दिइएको वृत्ताकार चित्रको अवलोकन गरी तलका प्रश्नहरूको उत्तर दिनुहोस्:

Figure: Concentric Circles with perpendicular radii $R$ and $r$

Part (a)

What is the area of shaded part?

(छायाँ पारिएको भागको क्षेत्रफल कति हुन्छ?)

Part (b)

If $R = 5\text{ cm}$ and $r = 3\text{ cm}$, find the area of shaded part.

(यदि $R = 5\text{ cm}$ र $r = 3\text{ cm}$ भए, छायाँ पारिएको भागको क्षेत्रफल पत्ता लगाउनुहोस्।)

Core Masterclass

Rules To Be Followed for Factorization Ex 10.1.2

Follow these sequential steps systematically to factorize any algebraic expression effortlessly in CDC Class 8/9/10 Mathematics.

बीजीय अभिव्यञ्जक खण्डीकरण गर्दा पालना गर्नुपर्ने मुख्य ४ वटा क्रमिक नियमहरू

Rule 1: Take Common First in Factorization (साझा लिनुहोस्)

अभिव्यक्तिमा जतिसुकै पदहरू भए पनि सबैभन्दा पहिले साझा गुणनखण्ड (HCF) बाहिर निकाल्नुहोस्।

2 Terms (२ वटा पदहरू)

Take Common Factor

When an expression has 2 terms in CDC Factorization Ex 10.1.2, inspect if a numerical constant or variable exponent is present in both.

$$2x^2 – 8x = 2x(x – 4)$$

3 Terms (३ वटा पदहरू)

Take Common Factor

If all three terms share a common factor, extract it before applying mid-term splitting or perfect square rule.

$$3x^2 + 6x + 9 = 3(x^2 + 2x + 3)$$

4 Terms (४ वटा पदहरू)

Grouping Technique

Group the expression into pairs of two terms each, then factor out common terms from each group.

$$ax + ay + bx + by = (x+y)(a+b)$$

Rule 2

Difference of Squares Formula

$$a^2 – b^2 = (a – b)(a + b)$$

Applied strictly in CDC Class 9 Math Ex 10.1.2 when two terms are perfect squares separated by a minus sign.

वर्ग-अन्तरको सूत्र प्रयोग गरी खण्डीकरण गर्ने तरीका।

Rule 3

Perfect Square Rule

$$a^2 \pm 2ab + b^2 = (a \pm b)^2$$

If the first and last terms are positive perfect squares and the middle term equals $2ab$.

पूर्ण वर्ग बनाउने सूत्र $a^2 + 2ab + b^2 = (a+b)^2$।

Rule 4

Mid Term Splitting

$$ax^2 + bx + c \rightarrow \text{Find } p, q \text{ s.t. } p \cdot q = ac, p+q = b$$

For general quadratic trinomials, split the middle coefficient into two numbers whose product equals $a \times c$.

बीचको पदलाई दुई भागमा फुटाएर खण्डीकरण गर्ने विधि।

Frequently Asked Questions (FAQ)

CDC Class 8/9/10 Math Factorization Exercise 10.1.2 FAQ

Q1: What is the main formula used in CDC Class 9 Math Factorization Exercise 10.1.2?

In CDC Class 8, 9, and 10 Mathematics Factorization Exercise 10.1.2, the primary formula used is the Difference of Two Squares formula $a^2 – b^2 = (a – b)(a + b)$ after taking out any common factor.

Q2: How to factorize algebraic expressions in CDC Class 9 Math Ex 10.1.2?

To factorize in Factorization Exercise 10.1.2, first check if there is a common factor to extract (e.g., $5x^2 – 20y^2 = 5(x^2 – 4y^2)$) and then convert the terms into perfect square forms $(a)^2 – (b)^2$.

Q3: Where can I find complete solutions for Grade 8/9/10 Math Factorization Exercise 10.1.2 Nepal?

You can find all step-by-step CDC Mathematics Grade 8/9/10 Factorization Exercise 10.1.2 solutions with video guides and key rules right here on this page.

Understanding Algebraic Factorization in CDC Secondary School Mathematics Nepal

Factorization is one of the most foundational topics in secondary school algebra under the Curriculum Development Centre (CDC) Nepal syllabus. Whether you are studying in Class 8, Class 9, or Class 10, mastering CDC Class 9 Math Factorization Exercise 10.1.2 and breaking down complex polynomials into simpler irreducible linear or quadratic factors is essential for solving algebraic fractions, HCF/LCM, quadratic equations, and higher-degree polynomial equations.

Why is CDC Factorization Ex 10.1.2 Important?

Algebraic factorization simplifies complex equations into manageable terms. For instance, when simplifying algebraic fractions, factorizing both the numerator and the denominator allows students to cancel common terms quickly. Furthermore, in geometry, expressing areas of geometric shapes—such as squares, rectangles, or concentric circles in CDC Class 9 Math Factorization Ex 10.1.2—using difference of squares $a^2 – b^2 = (a-b)(a+b)$ provides clear formulas for real-world application.

Summary of Steps to Solve Any CDC Math Ex 10.1.2 Factorization Problem:

Check for Common Terms: Always inspect every term in the expression in CDC Math Factorization Ex 10.1.2. If there is a common coefficient or variable, factor it out first. Example: $5x^2 – 20y^2 = 5(x^2 – 4y^2)$.
Count the Terms:
- 2 Terms: Look for difference of two squares formula $a^2 – b^2 = (a-b)(a+b)$.
- 3 Terms: Check if it is a perfect square trinomial $a^2 \pm 2ab + b^2$, or apply mid-term splitting $ax^2 + bx + c$.
- 4 Terms: Rearrange terms and group them into pairs to factor out common binomials.
Verify irreducibility: Check if any of the resulting binomial factors in CDC Factorization Ex 10.1.2 can be further factorized.

Previous: Ex 10.1.1 (Taking Common)

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